#185931

order	512
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×#120
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2336 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6)(7,8,15,11)(9,12)(10,13)(14,16,20,18)(17,19)(21,22,25,29,26,30,24,28)(23,31,32,27,34,35,33,36);(2,4)(5,7)(6,11)(8,13)(9,14)(10,15)(12,18)(16,19)(17,20)(21,23)(22,27)(24,32)(25,33)(26,34)(28,35)(29,31)(30,36);(5,8,10,11)(6,7,13,15)(9,16,17,18)(12,14,19,20);(5,9,10,17)(6,12,13,19)(7,14,15,20)(8,16,11,18)(21,24,26,25)(22,28,30,29)(23,32,34,33)(27,35,36,31)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=hi,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=ei,f^b=fh,g^b=ghi,e^c=eh,g^c=gh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72266618
maximal normal subgroup of	#72266618
maximal quotient of	#72266602

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 93127);